what does an element of span(S) look like? more pointedly, how do you form a "linear combination of the vj" what does that MEAN?

when you have hit upon that, you will see at once that two such linear combinations may be combined, and that BY THEIR LINEARITY, combined into a single linear combination.

here is a two-dimensional example: suppose S = {u,v}. we may regard an element of span(S) as being of the form a1u + a2v. another such element might be: b1u + b2v.

what could be more natural than (a1u + a2v) + (b1u + b2v) = (a1+b1)u + (a2+b2)v. is this not a linear combination of u and v, as well?

extend this idea to a scalar product k(a1u + a2v).

it should be clear that if S is a non-empty set, that span(S) has "one or more vectors" in it. convince yourself that for u in S, ku in span(S) implies that the 0-vector is in span(S).

can you see that -u has to be in span(S) as well? what further convincing might you need to see that span(S) has all the properties of a vector space in and of itself, and being a subset, is thus a sub-space?